To this end, we first notice that the zero-energy dislocation modes \(|\Psi _0\rangle \)  in the parent first-order topological insulator are linear combinations of the zero modes living on the reconnected edges, across the antiphase boundary, perpendicular to the Burgers vector [Fig. REF (a)]. The edge modes are the eigenstates of the matrix \(A_{\bf b}=i\Gamma _{\bf b}\Gamma _{3}\) , where \(\Gamma _{\bf b}={\Gamma }\cdot {\bf b}/|{\bf b}|\) , and \(\Gamma _3\)  is the mass matrix for the first order topological state (the Jackiw-Rebbi mechanism [1]}). The crucial observation is that the HOT mass matrix \(\Gamma _4\)  commutes with the dislocation/edge-mode matrix \(A_{\bf b}\) , \([\Gamma _4,A_{\bf b}]=0\) . The HOT mass matrix therefore reduces in the eigen-subspaces of \(A_{\bf b}\) , symmetrically splitting otherwise degenerate zero modes. Furthermore, this implies that the modes do not change the form, i.e. they remain localized, after introducing the HOT mass. The energy splitting \(\delta E=\langle \Psi _0| \Delta [{\bf K}_{\rm inv}-i{\bf b}({\bf b}\cdot {\nabla })]|\Psi _0\rangle \ne 0\) , unless \(\Delta [{\bf K}_{\rm inv}-i{\bf b}({\bf b}\cdot {\nabla })]=0\) , i.e. when the HOT mass vanishes in the direction of the Burgers vector. Therefore, the interplay between the orientation of the HOT mass domain wall and the Burgers vector of the lattice defect pins the dislocation modes precisely at zero energy when they are parallel to each other. The mechanism for the splitting of the dislocation modes is also operative for gapping out the edges, ultimately yielding the corner modes, implying that a dislocation can directly probe higher-order bulk-boundary correspondence.
